An overview of spatial microscopic and accelerated kinetic Monte Carlo methods

Abstract

The microscopic spatial kinetic Monte Carlo (KMC) method has been employed extensively in materials modeling. In this review paper, we focus on different traditional and multiscale KMC algorithms, challenges associated with their implementation, and methods developed to overcome these challenges. In the first part of the paper, we compare the implementation and computational cost of the null-event and rejection-free microscopic KMC algorithms. A firmer and more general foundation of the null-event KMC algorithm is presented. Statistical equivalence between the null-event and rejection-free KMC algorithms is also demonstrated. Implementation and efficiency of various search and update algorithms, which are at the heart of all spatial KMC simulations, are outlined and compared via numerical examples. In the second half of the paper, we review various spatial and temporal multiscale KMC methods, namely, the coarse-grained Monte Carlo (CGMC), the stochastic singular perturbation approximation, and the τ -leap methods, introduced recently to overcome the disparity of length and time scales and the one-at-a time execution of events. The concepts of the CGMC and the τ -leap methods, stochastic closures, multigrid methods, error associated with coarse-graining, a posteriori error estimates for generating spatially adaptive coarse-grained lattices, and computational speed-up upon coarse-graining are illustrated through simple examples from crystal growth, defect dynamics, adsorption-desorption, surface diffusion, and phase transitions.

Ternary Fe–Cu–Ni many-body potential to model reactor pressure vessel steels: First validation by simulated thermal annealing

“…For a proper comparison to experiments, we should thus also increase the simulation box by at least an order of magnitude. Noting that the physical simulation time shows a quadratic increase with box size [54], it is clear that our AKMC tool in its current implementation is not adequate for direct comparison with experiments. Despite the limitations of the AKMC simulations, good qualitative agreement with experiments was found.…”

Crossing the mesoscale no-man<U+2019>s land via parallel kinetic Monte Carlo.

“…The relaxation time for the longest mode can be estimated by fitting the dependence of the amplitude on time to an exponential function. Figure 35 shows this fit for various system sizes L; the straight lines on this log-log plot indicate an exponential decay as predicted by the continuum theory in equation (16). The relaxation time τ for each system is given by the negative reciprocal of the slope of the line.…”

“…Two additional recent papers which also give nice descriptions of KMC and related MC methods are [16,83].…”

Coarse-graining Calcium Dynamics on Stochastic Reaction-diffusion Lattice Model

“…The method is efficient in describing much larger length scales than conventional MC simulations while still incorporating microscopic details, and resulting in significant computational savings. An overview of the method is given in [16].…”